### Abstract

Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

Original language | English (US) |
---|---|

Pages (from-to) | 1-108 |

Number of pages | 108 |

Journal | Periodica Mathematica Hungarica |

Volume | 70 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

Externally published | Yes |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Central limit theorem
- Discrepancy
- Lattice point counting in specified regions
- Law of the iterated logarithm

### Cite this

*Periodica Mathematica Hungarica*,

*70*(1), 1-108. https://doi.org/10.1007/s10998-014-0064-x

}

*Periodica Mathematica Hungarica*, vol. 70, no. 1, pp. 1-108. https://doi.org/10.1007/s10998-014-0064-x

**Pell equation and randomness.** / Beck, József.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Pell equation and randomness

AU - Beck, József

PY - 2015/1/1

Y1 - 2015/1/1

N2 - Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

AB - Finding the integer solutions of a Pell equation is equivalent to finding the integer lattice points in a long and narrow tilted hyperbolic region, located along a straight line passing through the origin with a quadratic irrational slope. The case of inhomogeneous Pell inequalities it is equivalent to finding the integer lattice points in translated copies of a long and narrow tilted hyperbolic region with a quadratic irrational slope. We prove randomness in these natural lattice point counting problems. We have two main results: a central limit theorem (Theorem 2.1) and a law of the iterated logarithm (Theorem 2.2). The classical Circle Problem (counting lattice points in large circles) is a notoriously difficult open problem. What our results show is that the hyperbolic analog of the Circle Problem is solvable with striking precision.

KW - Central limit theorem

KW - Discrepancy

KW - Lattice point counting in specified regions

KW - Law of the iterated logarithm

UR - http://www.scopus.com/inward/record.url?scp=84925501486&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925501486&partnerID=8YFLogxK

U2 - 10.1007/s10998-014-0064-x

DO - 10.1007/s10998-014-0064-x

M3 - Article

AN - SCOPUS:84925501486

VL - 70

SP - 1

EP - 108

JO - Periodica Mathematica Hungarica

JF - Periodica Mathematica Hungarica

SN - 0031-5303

IS - 1

ER -